No, they do not
Golems are created using very similar methods of creating magic items, but are not magic items themselves once created. They are creatures, instead, of type construct.
The Craft Construct feat allows a spellcaster to create all manner of permanent constructs in a process much like magic item creation.
There is absolutely no reference in the rules stating that constructs, or golems more specifically, have a magic aura that could be detected. They are magical creatures, just like familiars and other magical beasts, but that doesn't mean they emanate a magical aura, which is what Detect Magic actually cares about. Not even the higher level version of the spell, Arcane Sight, will actually detect creatures as magical, but the magical auras affecting them and magical items they are carrying.
We also have clear exceptions to this, as Animated Objects if created using temporary magical effects (Animate Objects or Permanency) are stated to be susceptible to dispelling and anti-magic, while permanent constructs are not:
Not all constructs are built with the Craft Construct feat. Spells like animate objects allow a caster to temporarily animate an existing object. These constructs are in many ways weaker than manufactured constructs, as they are susceptible to dispelling and antimagic.
A caster can use the animate objects spell to instantly create a temporary construct. A permanency spell cast upon an Animated Object makes the construct permanent; however, it can still be dispelled or suppressed by antimagic. Craft Construct creates permanent animated objects not susceptible to dispelling and antimagic.
However, this doesn't mean that a specific golem cannot be detected if they have a supernatural ability like an aura (like a Cryptguard), or a constant magic effect (like the Graven Guardians). They are also identifiable by Knowledge (Arcana), and there are specific constructs that are made for the purpose of looking like a statue, like the Caryatid Column, Guardian Gargoyles and Cephalophore.
This has been confirmed by James Jacobs (Creative Director) on his Ask me anything thread:
What happens when a PC 'scans' a magical beast with detect magic?
Unless a monster specifically has a special quality that says otherwise (and I'm pretty sure there are none that do), you can't detect ANY monster (including magical beasts, undead, or constructs) with detect magic. Creatures don't have magic auras. Spell effects on them do, though.