No, there is no free equipment when you multiclass unless explicitly stated.
In the PHB it states:
"When you gain a new level in a class, you get its features for that level. You don't, however, receive the class's starting equipment, and a few features have additional rules when you're multiclassing: Channel Divinity, Extra Attack, Unarmored Defense, and Spellcasting."
Think of it this way; if you started as a Warlock then multiclassed into a Paladin, would they suddenly find platemail under their pillow?
On page 148 of the PHB, under Adventuring gear is a list of items, including Arcane Focus, which run between 5-20 gp. And there is nothing in the description of Warlock that would suggest that the focus needs to be tied to your specific patron.