After Some Time my Serial Port Stops Refreshing

Try this....

do {
digitalWrite(relay1, HIGH);
Serial.println("in do loop");
} while (((V1HON) != (myRTC.hours)) && (V1MON) != ((myRTC.minutes)));

if (((V1HON) == (myRTC.hours)) && (V1MON) == ((myRTC.minutes))) {
Serial.println("in relay loop");
Relay1();
}
}