In this case, there are two ordinal (ordered categorical) variables, each with a different number of levels or categories.
Probably the most recommended method to determine the association for this case is Kendall's tau-c. This statistic, and the associated test, is designed to handle the fact that there are a different number of levels in each of the variables.
Kendall's tau-b is similar, but is more appropriate for cases where the two ordinal variables each have the same number of levels. However, this statistic is more commonly available in software.
Spearman's correlation, as long as it is an implementation that handles ties in values, is also a reasonable option for ordinal variables.
In reality, these three correlation methods typically give similar results. Practical considerations such as which methods are available in software or which methods will be familiar to the audience may be the deciding factor.
Two other measures of association for ordinal variables are Somers’ D (or delta) and Goodman and Kruskal's gamma. Somers' D designates one variable as the independent variable and one as the dependent variable.
Polychoric correlation is used for ordinal variables based on the assumption that the ordinal variables represent latent, normally-distributed variables.