May and June are quiet for visual meteor hunters as the majority of showers are ‘daytime’ events and their radiants are too close to the Sun for us to observer visually. However if you have a radio detector you should pick them up. There’s more information about how to make a radio detector here and more on the daytime showers here.
The Eta Aquariids (ETA) will be largely washed out by the Full Moon on the 7th of May as will the Eta Lyrids (ELY) on the 9th. Given the moonlight the hourly rates for these showers are likely to be 2-3 combined.
The June Bootids (JBO) are listed in most catalogues as peaking on the 27th of June but have shown no significant activity since 2004. Nothing much is expected in 2020 either, but if you do see any activity please let me know as it would be scientifically interesting as the parent body comet 7P/Pons-Winnecke has changed orbit since 2004.
There’s one other possible interesting event: on the 14th of May, debris from Apollo asteroid 461852 may pass close enough to the Earth to give rise to slow-moving meteors. If you are out and about that night and see anything please do report it.