Center Controller reported an aircraft flew below the MVA by 1;300 feet near mountain side terrain.

Narrative

I was working a session with plenty of weather deviations with the cargo arrival push of the evening. Instructed Aircraft X on initial call to depart ZZZZZ intersection on a heading of 290 degrees to clear a mountainous terrain south of their position. Minutes later; I observed Aircraft X overflying the mountainside terrain below a MVA of 5;700 at 4;500 feet. I immediately advise Aircraft X to turn northbound on a heading of 310 to clear the terrain and the aircraft proceeded clear of the terrain without further incident.The RX/RXX sector was aware of the weather complexity in my sector; but was not giving me the traffic in an orderly and properly spaced manner. I asked the CIC to assist me with coordinating other sectors the departure headings due to the weather deviations; but the RX/ RXX sector was not asked to help me out with the situation by providing sufficient spacing.Also; Aircraft X did not comply with my initial instructions and due to the constant workload; I did not see this situation developing.