There seemed to be several slightly different notions of convergence spaces that were considered extensively in the 70ies. So I apologize if the following references do not actually answer your question (because I am missing some subtle differences between the definitions used in the papers below and the definition you linked to).
For the convergence spaces of Cook and Fischer, Theorem 5 in the following paper should prove that they are cartesian closed:
- H.R. Fischer and C.H. Cook. On Equicontinuity and Continuous Convergence. Math. Ann. 159 (1965), pp. 94-104. (available here)
For a slightly different notion of convergence, the $L\ast$-spaces, there is the following paper:
- G.A. Edgar. A Cartesian closed category for topology. General Topology and Appl. 6 (1976), no. 1, 65–72. (available here)
There are also uniform convergence spaces, cartesian closed by the following paper:
- R.S. Lee. The category of uniform convergence spaces is Cartesian closed. Bull. Austral. Math. Soc. 15 (1976), no. 3, 461–465, DOI: 10.1017/S0004972700022905.