Step 1: Identify the fixed expression.
The sentence contains the phrase
\[
\boxed{\text{come into effect}}
\]
which is a standard English expression.
Step 2: Recall its meaning.
The phrase come into effect means
• to become operative,
• to start being applied, or
• to begin functioning officially.
Since the policy begins to operate from the first day of the next month, the correct preposition is into.
Step 3: Analyse the options.
• (A) to -- Incorrect. "Come to effect" is not a correct expression.
• (B) into -- Correct. "Come into effect" is the standard idiomatic expression.
• (C) in -- Incorrect. "Come in effect" is grammatically incorrect.
• (D) onto -- Incorrect. It does not fit the context.
Step 4: Final conclusion.
\[
\boxed{\text{The new office policy will come into effect from the first day of the next month.}}
\]
Hence, the correct option is \(\boxed{(B)}\).