Probably yes, but she has more favor for inorganic constructs.
There's a chapter about Brigh in Inner Sea Faiths, and another chapter in Lords of Rust, the 2nd book of the Iron Gods adventure path. Nothing in either chapter suggests that inorganic constructs are outside of the Brigh relgion.
Superficially, Brigh is the patron goddess of clockwork, machines, and metalwork. There is lots of evidence for this.
- Brigh's alternate name is the "Whisper in the Bronze".
- She is often depicted as "a slender human woman made
of bronze clockwork".
- According to Lords of Rust, priests of Brigh can add the Clockwork Spy, Clockwork Servant, and Clockwork Soldier to the list of summonable creatures via Summon Monster.
- The head priest at a Brigh temple is called "High Clockmother or High
However, whenever these sources mention how Brigh cares for constructs, it never specifically says that the constructs must be mechanical or inorganic. There is plenty of evidence that Brigh is the patron goddess of all constructs, even if the materials themselves are organic.
According to Inner Sea Faiths, Brigh is described as "a patron of all invention". That's a large category, and should include both organic and inorganic inventions.
Clerics of Brigh can prepare Unbreakable Construct and Control Contruct (page 32). Both of those spells have a target of "one construct", and nothing about the spell text indicates that the target must be metallic.
I only found one mention of flesh golems, located in the "Planar Allies" subsection (page 33):
Many of Brigh’s divine servants were once mortal inventors, but have been recreated in “perfect” mechanical forms; she treats them as her children or younger siblings. Her following servitors are the ones most likely to respond
to planar ally.
Karapek (unique human alchemist): This dark-haired man is made of bronze, and has large hands and a back reinforced with struts of polished steel. He is knowledgeable regarding constructs and flesh golem construction, and
sees himself as an ambassador between living creatures and sentient constructs.
First, note the word "perfect" when describing inorganic constructs. This strongly suggests that Brigh favors mechanical creatures above others.
Still, it's plausible that if one of Brigh's divine servants is known for overseeing flesh golems (and similar alchemical constructs), then those types of constructs should also be within the realm of Brigh worship.