No, Shakespeare seems to have had a much more mundane inspiration- he copied someone. Several people in fact.

The earliest version of the tale we came to know as Romeo and Juliet was seemingly written by the Piedmontese cleric and writer of earthy tales, Matteo Bandelo, Bishop of Agen. It was he who published a large collection of tales, Novelle, in 1554. In these contains what is genuinely believed to be the first version of Romeo and Juliet.

Supposedly these were translated into French by the scholar Pierre Boaistuau and then into English. The earliest version of Romeo and Juliet in English was in in 1562 by the poet Arthur Brooke and 5 years later 1567 William Painter published his prose version of the tale.

Painter seems to have used several tales of Bandelo in his work and it is clear that his book The Palace of Pleasure seems to have inspired several of Shakespeares plays, including All’s Well That Ends Well, Timon of Athens, and Edward III. Meanwhile it appears Shakespeare was influenced directly by Bandelo for the plays Much Ado About Nothing, 12th Night and Cymberline.

Shakespeare didn’t copy the tales verbatim; indeed he specialised in lifting stories from the page and making them dynamic and more engaging on the stage; he would create new characters, change whole sections, introduce changes in time and pace, all focused on entertaining his audience.

With it’s opening monologue presenting the death of the lovers as a foregone conclusion (deliberately removing any vestige of hope for them and as such creating a perfect tragedy), followed swiftly by a series of ribald and rather crude bawdy jokes and then just as quickly by a pitched battle in stage, Romeo and Juliet is an example of Shakespeare displaying his first flowering’s of truth mastery of stagecraft.

He would grab the audience quickly and not let go.

Now the idea of ‘star crossed lovers’ or doomed love is a popular tradition in European literature, but given the specificity of the question, then no, no direct link to the Cleopatra story is to be found.

If you are interested, you can compare the versions below: https://archive.org/details/cu31924102029083/page/n195 Bandelo’s verison; this is a 19th century translation into English; the Romeo and Juliet story stars on page 196.

https://shakespeare-navigators.com/romeo/BrookeIndex.html A rather excellent guide to Brooke’s poem Romeus and Juliet, and a full transcript of it (broken into digestible chunks).

http://www.gutenberg.org/files/34840/34840-h/34840-h.htm#page80 William Painter’s Palace of Pleasure, containing on page 80, the tale of ‘Rhomeo and Ivliette’.