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You did not tell who is a, so let me chose something "random". Is the following satisfactory:

sage: p = 7
sage: r = 3
sage: q = p^r
sage: F = GF(p)
sage: a = F(2)

sage: R.<x> = PolynomialRing(F)
sage: P = R(cyclotomic_polynomial(r))
sage: P
x^2 + x + 1
sage: P.parent()
Univariate Polynomial Ring in x over Finite Field of size 7
sage: y = P.roots()[0][0]
sage: y
4
sage: y.parent()
Finite Field of size 7
sage: P(y)
0
sage: Q = (x-y)^2-a
sage: R = x^((q-1)/2)-1
sage: gcd(Q, R)
x + 6