What could be the correct idiom for expressing that someone is baking up false allegations without evidence?

An idiomatic verb phrase that refers to the act of concocting false allegations is "to trump up." As in,

The child trumped up charges of domestic abuse against her mother for making asparagus for dinner.

You could also use "trump up" as an adjective describing the type of allegations.

The child's trumped up allegations were deemed selfish and were ultimately dismissed.

trump up: to devise deceitfully or dishonestly, as an accusation; fabricate.

As noted in comments, “cooking up false allegations” and “barking up the wrong tree” are more-idiomatic forms of two expressions mentioned. But note (a) “making up false allegations” probably is more commonly heard than cooking them up, and (b) “barking up the wrong tree” does not refer to making up false allegations, it means to be off on a wrong track; for example, dogs barking at a tree with no racoon in it.

An interesting figurative expression is “made of whole cloth”. From wiktionary:

• (figuratively, used attributively or preceded by various prepositions) The fictitious material from which complete fabrications, lies with no basis in truth, are made. [eg] Mr. Doe's account of the accident was made from whole cloth.
• Something made completely new, with no history, and not based on anything else. [eg] The plans for the widget were drawn from whole cloth.