This is a fun observation... but I think you have a mistake. I wrote a quick python script to generate the fibonacci numbers and primes and make the counts and this is what I get:

Between 5 and 8 there is 1 prime: 7

Between 8 and 13 there is 1 prime: 11

Between 13 and 21 there are 2 primes: 17, 19

Between 21 and 34 there are 3 primes: 23, 29, 31

Between 34 and 55 there are 5 primes: 37, 41, 43, 47, 53

**Between 55 and 89 there are 7 primes: 59, 61, 67, 71, 73, 79, 83**

Between 89 and 144 there are 10 primes: 97, 101, 103, 107, 109, 113, 127, 131, 137, 139

Between 144 and 233 there are 16 primes: 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Between 233 and 377 there are 23 primes: 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373

Between 377 and 610 there are 37 primes: 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607