In this study, our aim is to investigate a new special number family, which is called the 3-parameter generalized quaternions with generalized Tetranacci numbers components with both non-negative and negative subscripts. According to this aim, we bring together the 3-parameter generalized quaternions (shortly 3PGQs), which are a general form of the quaternion algebra according to 3-parameters, and generalized Tetranacci numbers, which are also the most general form of all of the fourth-order recurrence sequences. Also, we examine some special cases of them and construct both new and fundamental properties such as recurrence relations, Binet formulas, generating functions, exponential generating functions, Poisson generating functions, summation formulas, matrix formulas, and special determinant equations. Furthermore, we give determinants, characteristic polynomials, characteristic equations, eigenvalues, and eigenvectors concerning the matrix representation of 3PGQs with generalized Tetranacci numbers components. Then, we examine a special case with respect to the given arguments and present an example.