In the US there is only one stock market (ignoring penny stocks) and handfuls of different exchanges behind it. NYSE and NASDAQ are two different exchanges, but all the products you can buy on one can also be bought on the other; i.e. they are all the same market. So a US equities broker cannot possibly restrict access to any "markets" in the US because there is only one.
(Interestingly, it is commonplace for US equity brokers to cheat their customers by using only exchanges where they -- the brokers -- get the best deals, even if it means your order is not executed as quickly or cheaply as possible. This is called
payment for order flow and unfortunately will probably take an Act of Congress to stop.)
Some very large brokers will have trading access to popular equity markets in other countries (Toronto Stock Exchange, Mexico Stock Exchange, London Stock Exchange) and can support your trades there. However, at many brokers or in less popular foreign markets this is usually not the case; to trade in the average foreign country you typically must open an account with a broker in that country.