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Yes, Zotero can be set up to pull in citation counts for publications.
Google Scholar
Any new items added to Zotero will automatically import the most recent citation counts from Google Scholar. These counts will appear in the Extra field.
For already existing items in Zotero, you can right click on My Library and select Update citations and Zotero will add Google Scholar citation counts to all items when available. You can also do this for individual collections or items.
CrossRef, Inspire HEP, & Semantic Scholar
Citation counts will not automatically be added with new items. Counts for CrossRef, Inspire HEP, & Semantic Scholar need to be added manually by right clicking on each publication and selecting each source. Results, if any, will appear in the Extra field.
For all resources to have the most recent count you will need to periodically right click on the collections or items and select Update Citations or manage Citation Counts.