The adaptive estimation of coexisting temporal vertex (node) and edge signals on graphs is a critical task when a change in edge signals influences the temporal dynamics of the vertex signals. However, the current Graph Signal Processing algorithms mostly consider only the signals existing on the graph vertices and have neglected the fact that signals can reside on the edges. We propose an Adaptive Joint Vertex-Edge Estimation (AJVEE) algorithm for jointly estimating time-varying vertex and edge signals through a time-varying regression, incorporating both vertex signal filtering and edge signal filtering. Accompanying AJVEE is a newly proposed Adaptive Least Mean Square procedure based on the Hodge Laplacian (ALMS-Hodge), which is inspired by classical adaptive filters combining simplicial filtering and simplicial regression. AJVEE is able to operate jointly on the vertices and edges by merging two ALMS-Hodge specified on the vertices and edges into a unified formulation. A more generalized case extending AJVEE beyond the vertices and edges is being discussed. Experimenting on real-world traffic networks and population mobility networks, we have confirmed that our proposed AJVEE algorithm could accurately and jointly track time-varying vertex and edge signals on graphs.