See this example (the bold quote is embedded mid-sentence...no colon precedes it and the period is outside the closing quotation-mark....but the word "State" actually begins a sentence in the source text):
A 7 November 2022 CNN piece says that “State Department spokesperson Ned Price said that any diplomatic solution needs to be worked out by Ukraine and Russia”.
Hi, Andrew—The convention is to use brackets around the first letter and make it lower case, the brackets showing that the letter in the original is not the same (case). The convention can also apply in reverse (bracketing an upper-case letter), an embedded sentence being represented as a separate sentence.
However, in your particular example, you need not even resort to using brackets. "Department" is capitalized in the quotation. This indicates that "State" must also be capitalized. It is highly unlikely that CNN would write "state Department spokesperson."
