23 × 32 × 11
1, 2, 3, 4, 6, 8, 9, 11, 12, 18, 22, 24, 33, 36, 44, 66, 72, 88, 99, 132, 198, 264, 396 and 792
No.
Its nearest Fibonacci number neighbors are 610 and 987.