Should my site be showing up in Google search results or am I being impatient?

I own a PI law firm in a small/medium sized city that opened about six weeks ago. I am using a digital marketing company that holds itself out as a one-stop shop — does web design, SEO, Google Ads, all of it.

My website has been online for six weeks. It has been indexed by Google. However, it does not appear in ANY search results that would be commonly searched. Am I wrong to be concerned about this? Does it really take several months for a website to even show up in Google search results?

I’m not asking for the website to rank on the first page immediately — I’m not insane. But it does seem concerning that despite being indexed, you can’t find it unless you are specifically searching for the name of my firm.