It appears from a computer exploration that (2,2) is the only solution.

Beginnings of a proof:

Consider n^5 + n^4 + 1 = 7^m

The LHS can be factored:

(n^2 + n + 1) * (n^3 - n + 1 ) = 7^m

Whenever n is 2 mod 7, both polynomials on the left are 0 mod 7, meaning they are multiples of 7.  But I have not found a way to prove that n=2 is the only value that makes each polynomial a power of 7.

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import math

for n in range(1,10000):

    lhs = n**5 + n**4

    m_precis = math.log(lhs,7)

    

    m1 = int(m_precis)

    m2 = m1 + 1

    

    rhs1 = 7**m1 - 1

    rhs2 = 7**m2 - 1

    if lhs == rhs1:

        print(n,m1)

    if lhs == rhs2:

        print(n,m2)