Do inter-dealer transactions increase the trade volume when I buy shares?

In modern stock exchange there is no dealer A and B. You place your order via broker and it matches to the order placed by other parties via their broker. So there is a single transaction of 100 shares

Suppose there are 4 entities involved in a transaction: me, my broker, dealer A, and dealer B.

That is a wrong assumption. The BROKER is not involved in the transaction - it is merely handling the paperwork and holding the shares in your name. The transaction is between you and the other side, totalling 100 shares. Why do you assume 2 dealers? EVERY transaction has ONE buyer (or seller) and the other side has as many participants as are needed to fill the order (in case of large orders).

Dealer A buys 100 shares from dealer B. Then, dealer A sells 100 shares to me via my broker.

This is not how it works. Your broker pushes the order on the exchange where A counterparty picks it up. Why would you assume a dealer buying from ANOTHER dealer?

If you buy from a market maker and they do not have the shares in their inventory, the volume will increase by a multiple of your trade size. These examples provided by Nasdaq illustrate this and how volume gets counted.