In this article, we define the Euler-Fibonacci numbers, polynomials and their exponential generating function. Several relations are established involving the Bernoulli F-polynomials, the Euler-Fibonacci numbers and the Euler-Fibonacci polynomials. A new exponential generating function is obtained for the Bernoulli F-polynomials. Also, we describe the Fibo-Bernoulli matrix, the Fibo-Euler matrix and the Fibo-Euler polynomial matrix by using the Bernoulli F-polynomials, the Euler-Fibonacci numbers and the Euler-Fibonacci polynomials, respectively. Factorization of the Fibo-Bernoulli matrix is obtained by using the generalized Fibo-Pascal matrix and a special matrix whose entries are the Bernoulli-Fibonacci numbers. The inverse of the Fibo-Bernoulli matrix is also found.