Beckham boys’ busy summer

BROOKLYN Beckham — the 18-year-old son of David and Victoria whom Page Six reported is dating YouTube star Madison Beer — is looking for an apartment in New York, we hear. Brooklyn — who just released a photography book, “What I See” — is “looking to take a bite [out] of the Big Apple,” a source dishes. “He’s searching for a space of his own in New York. He’s planning on moving there next month to pursue his photography career, among other things.” But the celebrity scion will be bicoastal “to pursue his relationship with Madison.” Beer was discovered by Justin Bieber online in 2012. Meanwhile, we hear that Brooklyn’s little brother Cruz, 12, is in LA working on an album as a follow-up to the Christmas single he dropped last year. A source tells us that Cruz has “recorded some covers, including [of ] Mumford & Sons, and is working on writing original material with Stephen Puth, brother of Charlie Puth. Charlie will come for a few sessions to accompany on piano.” But a rep insisted Cruz has just “been working on his guitar lessons during his LA holidays.”