DD#1 stopped taking naps at age 2.
I've got two words for you...
DD#1 is now 3 years, 3 months. and she has quiet time everyday after lunch in her room - on her bed, with quiet toys and music and books to look at. She stays in there for 20-30 minutes on her own. I can sometimes encourage her to stay longer. I never, ever force her, though. Quiet time is something I want her to enjoy and appreciate.
Also, you can also work on teaching your DS that YOU need "quiet time", too. I taught my DD#1 that Mommy needs quiet time in the early morning. I taught her was 7am looks like on the clock, and she knows that if she wakes up early, she can turn on her lights and play until 7am. Then I will go in and snuggle with her. My quiet time involves me staggering out of bed, getting a cup of tea and some cereal, reading the news a bit. I am so much more ready to greet the girls in the AM, if I can have my quiet time, too!
Your DS may not be able to read a clock well enough yet, but you can find other times in the day to start teaching him about "Mommy's Quiet Time". Find some times in the day where there is a natural break in the routine/rhythm of the day, where he may be in a quiet mode himself, and work in those quiet times for yourself.
edited to add - I know you said he doesn't want quiet time
but, he doesn't have to lay still on his bed to have quiet time. He can just quietly play in his room with some quiet toys, books, etc. Some nice music. I bet if you started it slowly, like 10 minutes a day, and let him get comfortable with the idea, he might like it! Maybe!
Seriously, I did NOT think my DD#1 was going to go for quiet time. So my sister suggested just 10 minutes a day - setting a timer for her, so she knew when quiet time was done. Well, after about 4 days, she started saying, "I'm going to stay in quiet time just a wittle (little) bit longer, ok?"
And now she is so used to it and likes it.