Model takes in QC after the first quiet drink (9)

'drink' is the definition.
(milkshake is a kind of drink)

'model takes in qc after the first quiet' is the wordplay.
'model' becomes 'make' (eg a make or model of car).
'takes in' means one lot of letters goes inside another.
'qc' becomes 'silk' (informal term for a Queen's Counsel barrister).
'after the first' means to remove the first letter.
'quiet' becomes 'sh'.
'silk' with its initial letter taken away is 'ilk'.
'ilk'+'sh'='ilksh'
'make' placed around 'ilksh' is 'MILKSHAKE'.