High-fidelity　quantum　state　transfer　is　critical　for　quantum　communication　and　scalable　quantum　computation.　Current　quantum　state　transfer　algorithms　on　the　complete　bipartite　graph,　which　are　based　on　discrete-time　quantum　walk　search　algorithms,　suffer　from　low　fidelity　in　some　cases.　To　solve　this　problem,　in　this　paper　we　propose　a　two-stage　quantum　state　transfer　algorithm　on　the　complete　bipartite　graph.　The　algorithm　is　achieved　by　the　generalized　Grover　walk　with　one　marked　vertex.　The　generalized　Grover　walk’s　coin　operators　and　the　query　oracles　are　both　parametric　unitary　matrices,　which　are　designed　flexibly　based　on　the　positions　of　the　sender　and　receiver　and　the　size　of　the　complete　bipartite　graph.　We　prove　that　the　fidelity　of　the　algorithm　is　greater　than　1-2ϵ1-ϵ2-22ϵ1ϵ2　or　1-(2+22)ϵ1-ϵ2-(2+22)ϵ1ϵ2　for　any　adjustable　parameters　ϵ1　and　ϵ2　when　the　sender　and　receiver　are　in　the　same　partition　or　different　partitions　of　the　complete　bipartite　graph.　The　algorithm　provides　a　novel　approach　to　achieve　high-fidelity　quantum　state　transfer　on　the　complete　bipartite　graph　in　any　case,　which　will　offer　potential　applications　for　quantum　information　processing.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature.