As a popular and practical query type in location-based services, the spatial keyword query has been extensively studied in both academia and industry. Meanwhile, with the growing demand for data privacy, many privacy-preserving spatial keyword query schemes have been proposed to deal with queries over encrypted data. However, none of the existing schemes preserve access pattern privacy, and the recent research illustrates that leaking such privacy may incur inference attacks and thus disclose sensitive information. In addition, most existing schemes only consider the boolean keyword search, which is not quite practical and flexible in real-world applications. To address the above issues, in this paper, we propose two privacy-preserving spatial keyword similarity query schemes that can preserve full and partial access pattern privacy, respectively. First, we present a basic privacy-preserving spatial keyword similarity query scheme (PPSKS) by integrating a secure set membership test (SSMT) technique with secure circuits. After that, to improve performance, we propose a tree-based scheme (PPSKS+) by employing a new index called FR-tree together with a predicate encryption technique that can encrypt FR-tree. Formal security analysis shows that: i) our proposed schemes can protect outsourced data, query requests, and query results; ii) our PPSKS scheme can hide full access patterns, while the PPSKS+ scheme preserves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> -access pattern privacy. Extensive experiments are also conducted, and the results indicate that our tree-based PPSKS+ scheme is much more efficient, almost two orders of magnitude better than our linear search PPSKS scheme in performing queries.