This paper studies an asymptotics of functional linear quantile regression in which the dependent variable is scalar while the covariate is a function. We apply a roughness regularization approach of a reproducing kernel Hilbert space framework. In the above circumstance, narrow convergence with respect to uniform convergence fails to hold, because of the strength of its topology. A new approach we propose to the lack-of-uniform-convergence is based on Mosco-convergence that is weaker topology than uniform convergence. By applying narrow convergence with respect to Mosco topology, we develop an infinite-dimensional version of the convexity argument and provide a proof of an asymptotic normality of argmin processes. Our new technique also provides the asymptotic confidence intervals and the generalized likelihood ratio hypothesis testing in fully nonparametric circumstance.