Time series clustering is essential in scientific applications, yet methods for functional time series, collections of infinite-dimensional curves treated as random elements in a Hilbert space, remain underdeveloped. This work presents clustering approaches for functional time series that combine the fuzzy $C$-medoids and fuzzy $C$-means procedures with a novel dissimilarity measure tailored for functional data. This dissimilarity is based on an extension of the quantile autocorrelation to the functional context. Our methods effectively groups time series with similar dependence structures, achieving high accuracy and computational efficiency in simulations. The practical utility of the approach is demonstrated through case studies on high-frequency financial stock data and multi-country age-specific mortality improvements.

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