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This article makes use of simultaneous decomposition of four quaternion matrixes to investigate some Sylvester-like quaternion matrix equation systems. We present some useful necessary and sufficient conditions for the consistency of the system of quaternion matrix equations in terms of the equivalence form and block matrixes. We also derive the general solution to the system according to the partition of the coefficient matrixes. As an application of the system, we present some practical necessary and sufficient conditions for the consistency of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>-Hermitian solution to the system of quaternion matrix equations in terms of the equivalence form and block matrixes. We also provide the general <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>-Hermitian solution to the system when the equation system is consistent. Moreover, we present some numerical examples to illustrate the availability of the results of this paper.