I'm inmigrant first year working in the US and I don't understand the quartely payments

I'm an immigrant, I'm legal, and it's my first year working. I started working as a medical interpreter as an independent contractor. The company asked me to register an EIN as a Sole Proprietor.

The company doesn't remove taxes from my payments.

First, I thought that I had to pay taxes by the end of the year with the 1099-NEC form my company will give me, but then I was looking that you have to make quarterly payments based on estimated payments (I'm really confused about this because I don't know what I will make during the year as my income isn't stable).

Basically, I already missed two quarterly payments (I started working in May).

Then I saw that:

You had no tax liability for the prior year. You are exempt from making the estimated quarterly payments based on estimates. Since it's my first year in the US, I'm exempt, right?

At this point, can I just wait until the end of the year when my company sends me the 1099 form?