@mmcdara I have attached a successful events version.
I will say, however, that the classical approach is simpler and real fast with stiff=true (i.e. rosenbrock).
The discrete variable f(t) used here is of type float and will take on 3 values: -1, 0, or 1.
Events are triggered by v(t) +eps =0, v(t)-eps = 0, and v(t)=0.
For both methods stiff=true is chosen although it probably has no effect in the events case.

 

pC and pE look the same and overlap totally in the last display with eps=1e-4.
With eps = 1e-3 computation time is much smaller and overlap is still good.
With eps = 1e-2 computation time is even smaller, but there is clear visual disagreement between pC and pE.
odeplot(SOL_E,[t,f(t)], T__0..2,numpoints=10000) will show you that an enormous amount of events takes place, so no wonder it takes time.

Download MaplePrimes20-11-07_classical_and_event.mw

@Muhammad Usman Commenting out what may interfere with the timing (such as remembering values from the plots) and omitting printing the loop I get 0.187s.
fnum can be seen afterwards simply by eval(fnum);
 

restart;

U := q^6*sin(p);
alpha := 1/3;
M1 := 3;
M2 := 3;
A:=Int(Int(U/((x^rho - p^rho)^alpha*(y^rho - q^rho)^alpha), p = 0 .. x), q = 0 .. y);

Q:=proc(r,xx,yy) evalf(eval(A,{'rho'=r,x=xx,y=yy})) end proc;
##plot3d(Q(1,x,y),x=0..1,y=0..1); # Test
## An animation in rho with only 6 frames
##plots:-animate(plot3d,[Q(rho,x,y),x=0..1,y=0..1],rho=1..6,frames=6);

## The revised loop:
t0:=time():
##printlevel:=3:
for i to 6 do
    for i11 to M1 do
        for j11 to M2 do 
            fnum[i, i11, j11] := Q(i,i11/M1,j11/M2)
        end do
    end do
end do:
time()-t0;
# time = 0.187s
eval(fnum);

@mmcdara I just tried these events:
 

events=[[[0,v(t)>0],b(t)=1],[[0,v(t)<0],b(t)=0]]

They seem to produce exactly what you have in your event version.

@mmcdara I would like to, but haven't seen a way out yet.

When I do ?ConvertIn the help page for modp1 opens.
On that page it says:
"The modp1 function accepts the following functions, whose names are also protected global names."

Then follows 5 colums of names some of which are in black (i.e. no link to help page). Among those is ConvertIn.
Its meaning is understood by modp1 and is described on that help page. In fact ConvertIn may just be nothing but a protected name used in a call to modp1 to tell it what to do.
Some of those with an actual link (the blue ones) like Add just brings you to the help page for add. That must be a mistake.
How Add is used is beyond me.

A common way of solving this kind of ode is to multiply by diff(y(x),x) on both sides and integrate like this:
 

restart;
ode:=2*diff(y(x),x$2)=exp(y(x));
IC:=y(0)=0,D(y)(0)=1;
diff(y(x),x)*ode;
ode1:=map(int,%,x)+(0=C1);
eval[recurse](convert(ode1,D),{x=0,IC});
## So C1 = 0.
ode2:=eval(ode1,C1=0);
odes:=solve(ode2,{diff(y(x),x)});
ode3:=op(odes[1]);
sol1:=dsolve(ode3);
eval[recurse](sol1,{x=0,IC});
## so _C1 = -2.
sol:=eval(sol1,_C1=-2);

Which one of these two (if any of them) did you mean:

ode1:=diff(U(x), x$2) + ((z+ I*y)^2/k1^2 -k2^2 + ((z+I*y)/k3)*sech(x)^2)* U(x);
ode2:=diff(U(x), x$2) + ((z+ I*y)^2/(k1^2 -k2^2) + ((z+I*y)/k3)*sech(x)^2)* U(x);

Maple 2020 solves both symbolically.

@Muhammad Usman 
1. You can make a very simple procedure (Q below) to return values of the integral for given x and y values.
2. In your original triple loop you didn't keep all values computed. fnum[i] was redefined for every new values of i11 and j11.

restart;

U := q^6*sin(p);
alpha := 1/3;
M1 := 3;
M2 := 3;
A:=Int(Int(U/((x^rho - p^rho)^alpha*(y^rho - q^rho)^alpha), p = 0 .. x), q = 0 .. y);
Q:=proc(r,xx,yy) evalf(eval(A,{'rho'=r,x=xx,y=yy})) end proc;
plot3d(Q(1,x,y),x=0..1,y=0..1); # Test
## An animation in rho with only 6 frames
plots:-animate(plot3d,[Q(rho,x,y),x=0..1,y=0..1],rho=1..6,frames=6);
## The revised loop:
printlevel:=3:
for i to 6 do
    #rho := i; #Not used
    for i11 to M1 do
        for j11 to M2 do 
            fnum[i, i11, j11] := Q(i,i11/M1,j11/M2)
        end do
    end do
end do;

@Muhammad Usman Could you please give the full code of what you tried to do. Perhaps upload a worksheet?
 

@Muhammad Usman Numerical integration requires x and y to be given numerical values.

So here is an example where x = 1 and y = 2:
 

for i from 1 to 6 do
   rho := i;
   fnum[i] := evalf(eval(Int(Int(U/((x^rho-p^rho)^alpha*(y^rho-q^rho)^alpha), p = 0 .. x), q = 0 .. y),{x=1,y=2}));
end do;

Compare that with:
 

evalf(eval([seq](f[i],i=1..6),{x=1,y=2}));

@Muhammad Usman I have revised my code above so U now is the revised U.
If you don't reverse the order of integration f[4] is not completed and other results are messier.

@Anthrazit By looking at the codes for the Standard and Simple versions of min (and by debugging both):
 

showstat(Units:-Standard:-min);
showstat(Units:-Simple:-min);
debug(Units:-Standard:-min);
debug(Units:-Simple:-min);

you will see that Units:-Simple:-min among several return statements has a return statement at the bottom 

return min(op(arguments))
where the local 'arguments' is just the list of arguments passed to Units:-Simple:-min.
What :-min gets as arguments is easily seen by doing debug(:-min);
So after a restart you can try:
 

restart;
min(15*Units:-Unit(m), 0);  # 0
max(15*Units:-Unit(m), 0); # 15*Units:-Unit(m)

@srikanthan It might help us if we knew the values of the parameters excepting of course R and thetaw, i.e.
{P, Pr, Q, Qc, S0, S1, S2, S4, c, lambda, n}.
Or at least an example of reasonable values.
Then we could make a procedure taking the value (Dth0) of D(theta)(0) and thetaw as input and R as output using as an extra boundary condition D(theta)(0) = Dth0.
 

If you insist on the initial conditions in dsys6 then it follows that n = 1. Simply use the connection between x and y given in the first equation together with the initial conditions.

I suppose then that there is some mistake here?

I tried (twice) asking for that helper library by writhing to physics@maplesoft.com , but I'm waiting for it to get to me.
Maybe I'm not patient enough?