Short answer. No, not for this. The attorney-client privilege has a built-in exception for a communication relevant to an issue of breach of duty by the lawyer to his or her client or by the client to his or her lawyer — a malpractice suit is exactly that kind of issue.
What the law says
Breach of duty by lawyer or client. As to a communication relevant to an issue of breach of duty by the lawyer to his or her client or by the client to his or her lawyer
Rule 130, Section 24 — Disqualification by reason of privileged communication[s]. Read the full provision →
The privilege itself is real and broad
The rule generally protects a client by preventing an attorney from being examined, without the client's consent, about any communication made by the client to him or her, or his or her advice given thereon in the course of, or with a view to, professional employment. This is a genuinely strong protection meant to let clients speak freely and openly with their lawyers, and it applies whether it is the lawyer, opposing counsel, or someone else entirely trying to get at those communications in another proceeding.
But the rule lists specific exceptions, and yours is one
The privilege is not entirely absolute. The rule carves out several named exceptions where the privilege does not apply, and one of them covers a communication relevant to an issue of breach of duty by the lawyer to his or her client or by the client to his or her lawyer. A malpractice claim is precisely and exactly an allegation of breach of duty by the lawyer toward his own client, so this exception speaks directly to your particular situation.
Why this exception exists
A privilege designed to protect the client cannot be turned around and used by the lawyer as a shield against the client's own claim that the lawyer failed in his duties. Letting the lawyer invoke privilege to hide the very communications relevant to whether he actually committed malpractice would defeat the whole purpose the privilege was created to serve in the first place — protecting the client, not shielding the lawyer from accountability.
The exception is limited to what is relevant to the breach
The exception opens the door specifically to communications relevant to an issue of breach of duty — it is not a blanket waiver of every confidential communication the two of you ever exchanged on unrelated matters. What becomes discoverable is what actually bears on whether your former lawyer breached a duty owed to you, not necessarily the entirety of the file covering every matter he ever handled for you.