Are there any good algebraic/algorithmic tools available to check if a given graph $H$ is a minor of $G$ from the adjacency matrix of $G$?
There is a general implementation in Sage. However, the algorithm runtime grows exponentially in the size of $H$. If you have a particular small $H$ in mind, there may be more efficient implementations available.
EDIT: In the comments below, the OP clarifies that the only case of interest is $H=K_5$. For this special case, the best known algorithm is due to Bruce Reed and Zhentao Li, "Optimization and Recognition for K5-minor Free Graphs in Linear Time," LATIN 2008, LNCS 4957, pp. 206–215, 2008. Reed's website also has a draft paper with fuller algorithmic details. Unfortunately, this algorithm is very complicated and I don't know if it has been implemented; I'd recommend emailing the authors to ask.
By the way, if you find yourself having to write your own code, there is an earlier and simpler (though asymptotically less efficient) algorithm due to P. J. McGuinness and A. E. Kezdy, "Sequential and parallel algorithms to find $K_5$ minor," SODA 1992, pp. 206–215.
A polynomial time algorithm for this is given in
Robertson + Seymour, Graph minors. XIII. The disjoint paths problem, Journal of Combinatorial Theory, Series B, 63 (1995), pp. 65–110.