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Essentially you are trying to compute the image of Frobenius endomorphism of 19*v^2 + 49*v + 8 in the field GF(67^3) with the modulus v^3+2. You can define it as

R.<v> = PolynomialRing(GF(67))
K.<w> = GF( 67^3, name='w', modulus=v^3 + 2 )
Frob = K.frobenius_endomorphism()
print(Frob)

to get

Frobenius endomorphism w |--> w^67 on Finite Field in w of size 67^3

Then it remains to compute Frob( 19*v^2 + 49*v + 8 ), which gives 15*w^2 + 4*w + 8.

Essentially you are trying to compute the image of Frobenius endomorphism of on 19*v^2 + 49*v + 8 in the field GF(67^3) with the modulus v^3+2v^3 + 2. (i.e. defined as $GF(67)[v]/\langle v^3+2 \rangle$). You can define it as

R.<v> = PolynomialRing(GF(67))
K.<w> = GF( 67^3, name='w', modulus=v^3 + 2 )
Frob = K.frobenius_endomorphism()
print(Frob)

to get

Frobenius endomorphism w |--> w^67 on Finite Field in w of size 67^3

Then it remains to compute Frob( 19*v^2 + 49*v + 8 ), which gives 15*w^2 + 4*w + 8.

Essentially you are trying to compute the image of Frobenius endomorphism on 19*v^2 + 49*v + 8 in the field GF(67^3) with the modulus v^3 + 2 (i.e. defined as $GF(67)[v]/\langle $\mathrm{GF}(67)[v]/\langle v^3+2 \rangle$). You can define it as

R.<v> = PolynomialRing(GF(67))
K.<w> = GF( 67^3, name='w', modulus=v^3 + 2 )
Frob = K.frobenius_endomorphism()
print(Frob)

to get

Frobenius endomorphism w |--> w^67 on Finite Field in w of size 67^3

Then it remains to compute Frob( 19*v^2 + 49*v + 8 ), which gives 15*w^2 + 4*w + 8.

Essentially you are trying to compute the image of Frobenius endomorphism on 19*v^2 + 49*v + 8 in the field GF(67^3)$\mathrm{GF}(67^3) \cong with the modulus v^3 + 2 (i.e. defined as $\mathrm{GF}(67)[v]/\langle \mathrm{GF}(67)[v]/\langle v^3+2 \rangle$). \rangle$. You can define it as

R.<v> = PolynomialRing(GF(67))
K.<w> = GF( 67^3, name='w', modulus=v^3 + 2 )
Frob = K.frobenius_endomorphism()
print(Frob)

to get

Frobenius endomorphism w |--> w^67 on Finite Field in w of size 67^3

Then it remains to compute Frob( 19*v^2 + 49*v + 8 ), which gives 15*w^2 + 4*w + 8.