In this paper, we consider the new family of recurrence sequences of (q,k)-generalized Fibonacci numbers. These sequences naturally extend the well-known sequences of k-generalized Fibonacci numbers and generalized k-order Pell numbers. We shall obtain a Binet-style formula and study the asymptotic behavior of dominant root of characteristic equation. Moreover, we shall prove some auxiliary results about these sequences. In particular, we characterize the first (q,k)-generalized Fibonacci numbers in terms of binary sequences.