Not without a lot of trouble.
First issue is that, even though the transcription has the option of identifying speakers (and you can edit the speaker designation etc), that information IS NOT transferred to captions created from the transcript. I assume they plan to do so, but I have seen no information as to the future intention.
Also, a sequence can only have a single active caption track. And the only Caption Track Styles apply to an entire track. So you can create a track for each speaker, and you can set a style to make the color different, but you can't show them at the same time. The exception (see below) is nesting and burning in.
See "Show Multiple Caption Tracks" in this thread:
https://community.adobe.com/t5/premiere-pro-discussions/faq-workaround-nesting-to-apply-effect-controls-to-caption-track/td-p/11935335
And a post in that thread re converting captions to graphics text:
https://community.adobe.com/t5/premiere-pro-discussions/faq-workaround-nesting-to-apply-effect-controls-to-caption-track/m-p/12090742#M350619
And a thread using caption "blocks":
https://community.adobe.com/t5/premiere-pro-beta-discussions/mulitple-captions-blocks-and-other-major-captions-issues/m-p/11902745#M2718
There are several relevant feature requests. Here's one:
https://adobe-video.uservoice.com/forums/911233-premiere-pro/suggestions/42970134-multiple-caption-tracks-visible-for-accessibility
Stan