Is it allowed to for two tables to have two types of relationships between them?
This might sound a bit crazy, but as noted by ypercube in the answer linked in the comments, it is best to consider this a separate relation: for which we need a new table Question_CorrectAnswer, and it's PK is QuestionID.
This means that we only need to update a single row to change the correct answer.
Our only issue is that to prevent update anomalies, whereby someone could enter an Answer as being the correct one for a different Question, we must redefine Answer's primary key as being AnswerID, QuestionID, and FK on those two columns together.
So we define our tables like this:
(please don't take this as the best way to write CREATE TABLE, I've left out constraint names for example)
CREATE TABLE Question (
    QuestionId int IDENTITY PRIMARY KEY,
    Title varchar(100) NOT NULL
)

CREATE TABLE Answer (
    AnswerId int IDENTITY,
    QuestionId int NOT NULL REFERENCES Question (QuestionID),
    Body varchar(1000) NOT NULL,
    PRIMARY KEY (AnswerID, QuestionID)
)

CREATE TABLE Question_CorrectAnswer (
    QuestionId int NOT NULL PRIMARY KEY,
    AnswerId int NOT NULL,
    FOREIGN KEY (AnswerId, QuestionID) REFERENCES Answer (AnswerID, QuestionID)
)

It does have the downside of requiring an extra join, though.