My wife menstruation start on June 2 if i have sex with her on June 23 until the day before her menstruation is safe enough unpregnancy?

That is a hard question to answer as ovulation is not an exact science. It depends on many factors including how many days your wife's menstrual cycle lasts. Does she get her period every 28 days? Every 30 days? or more?
A woman's fertile time is usually a five day period starting approximately 14 days after the first day of her period. So you should avoid having sex on those days. However, if she has irregular periods her fertile time can be hard to deterimine.
Some things she can do to track when she is fertile or ovulating is to take her temperature every morning when she wakes up. Just prior to ovulation her temperature will drop and then start to rise a tiny bit at a time.
She can also check her cervical mucous. As she gets closer to ovulation the mucous will thicken and sometimes resemble and eggwhite in consistency. This can be done by inserting a finger in her vagina and looking at the mucous.
Lastly, there are ovulation kits you can buy at the store that she can use to deterimine if she is ovulating. They resemble pregnancy tests and she urininates on a stick like test that will measure her FSH, the hormone that stimulates ovulation and increases right before and at ovulation. These are not exact science either and I dont recommend using ovulation tests and birth control.
If she does not wish to take birth control and you are nervous about determining when she is fertile and risking pregnancy, condoms are a good method you can use.
Hope this helps!
Kelly