Swansea all-but secured their Premier League status with a vital 4-1 win over Aston Villa at the Liberty Stadium and Williams wants interim manager Monk to be rewarded for all his hard work.
Monk, who took over on a temporary basis when Michael Laudrup was sacked in February, is expected to hold talks with the club about his future in the next few weeks.
Club captain Williams said: It's a massive relief for us and we are very happy with the performance and the win.
"I said from the first week when I saw his work with us that he deserves the job. We are pleased that we may have stayed up. He deserves the job, we are all enjoying it, training is good so I think he really deserves his chance.
The Wales captain, who has been linked with a summer move away from the club, says he is happy at Swansea and remains relaxed about his future.
The 29-year-old added: "I have another year left on my contract and I have never had a dispute with the chairman yet, so we are relaxed about that and it's nothing to be worried about."