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I am new to the Seurat package and was looking at the tutorials on the Seurat website and am just curious to know if there is any way to perform a t.test for the expression of a specific gene between two groups within the Seurat package, even possibly showing these results with box plots? Or any way to extract this information from the Seurat object in order to do this using ggplot or anything of that sort? I don't have any specific examples as I've just been going through the Seurat tutorials trying to understand everything, but if anyone has any sort of advice on the topic I would greatly appreciate it!

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Most of your questions are part of the Vignettes on the Seurat home page.
In the differential expression Vignette you can see that you can run Student's t-tests in the FindMarkers wrapper function.
However, simple t-tests may not be the best option for single cell data. There are many algorithms for DE testing with some being directly accessible through FindMarkers. The alogrithm implemented in MAST is recommended frequently.

Regarding the box plots you should have a look at the violin plot function of Seurat:

VlnPlot(SeuratData, c("gene_of_interest"), group.by = "groups_compared_in_DE_test")
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