Is it necessary that a Christian should know the exact date or time of his or her conversion?

No. There is no Scripture which even suggests that we must know or remember the date when they were saved. Many believers do remember the exact date when they were saved. Others, especially those saved when they were very young, cannot remember the date when they confessed their sins and received Jesus Christ as their Lord and Saviour. John 5:24 and many other Scriptures teach us that conversion takes place when we believe the testimony of God. What is important, then, is that on the authority of God's Word we have a present assurance that we are born of God and have eternal life through faith in Jesus Christ. The knowledge of the exact date is not necessary or vital to one's salvation or Christian testimony.