This short note introduces the idea of second-generation Mersenne primes. Starting from a known Mersenne prime P = M_p = 2^p - 1, we use P itself as a new prime exponent and consider the second-generation Mersenne number M_P = 2^P - 1. If M_P is prime, then the usual Mersenne divisor form q = 2PK + 1 applies with q = M_P itself. In that special case, the coefficient K is exactly the Fermat quotient in base 2: K = Q_P(2) = (2^(P-1) - 1) / P. Thus, M_P = 2P Q_P(2) + 1. The note applies this observation to the known Mersenne primes from rank 48 to rank 52, including the current record Mersenne prime M_136279841. The purpose is not to claim a new primality test, but to isolate a simple structural coordinate connecting Mersenne primes, second-generation exponents, and Fermat quotients.