Any idea about this doubt about order of element of a group.

let g belongs to a group G so if order of g^n is m , then is it necessary that order of g^m will be n.
if it will be, then for what types of m,n. or for all integers

no.

as a counter example, consider a cyclic group of order 6, with generator x.

consider the element x^4: it has order 3:

(x^4)^2 = x^8 = (x^6)(x^2) = x^2
(x^4)^3 = x^12 = (x^6)^2 = e.

but x^3 does not have order 4, it has order 2:

(x^3)^2 = x^6 = e.

in general, if |x| = k, then |x^n| = k/gcd(k,n).

in our example above we saw that for k = 6,

we have |x^4| = 6/gcd(4,6) = 6/2 = 3,

while |x^3| = 6/gcd(3,6) = 6/3 = 2.

but if |g| = mn, then:

|g^m| = mn/gcd(m,mn) = mn/m = n, and

|g^n| = mn/gcd(n.mn) = mn/n = m.