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We consider a $q$-Painlev\'e IV equation which is the $A_4^{(1)}$-surfacetype in the Sakai's classification. We find three distinct types of classicalsolutions with determinantal structures whose elements are basic hypergeometricfunctions. Two of them are expressed by ${}_2\varphi_1$ basic hypergeometricseries and the other is given by ${}_2\psi_2$ bilateral basic hypergeometricseries.

Hypergeometric Solutions of the $A_4^{(1)}$-Surface $q$-Painlev\'e IV Equation