Is it safe to wait 4 months for my kidney stone to be destroyed while having a double j?

See your urologist. The double j stent will keep urine flowing past an obstruction, but the stone will not be affected by it. Only your urologist can say if it needs lithotripsy or surgical extraction.
Kidney stones. The short answer is, "yes." the only significant complication that could occur would be obstruction with infection. So, if fever occurs, report it to your doctor.