How does one download investment holdings into quicken from the brokerage account and accept it?

I don't understand the "without attempting to match prices" statement in your question.

When you first download from a brokerage account a certain number of transactions are downloaded - 90 day's worth is common, but it could be more.  If the account had transactions before the download "window" then the security positions derived from those transactions might not agree to your actual holdings in the account.  But the initial download does include the correct number of shares of any securities currently held, and if the positions derived from the initial download don't agree to those correct numbers Quicken proposes a Placeholder to get the number of shares in Quicken correct, and also allows you to bring the total basis in each security in agreement with the broker.  (Of course the broker might not have the correct basis for each and every security, depending on the circumstances.)

You can of course simply accept the Placeholder without getting providing a correct adjustment figure for the basis.  At that point your number of shares are correct, the market value (# of shares x current quote) is correct, but your total basis in each security could be right or wrong.  And you move forward from there.

" I don’t understand why the number of shares in the holdings become different in Quicken."

So by "balances" you're speaking of the calculated Market Value, i.e., # of shares x quote? 

And you appear to be saying that at some point the # of shares per Quicken agrees to the broker, but then somehow Quicken's number of shares drifts away from the correct number of shares?

You are in the perfect position to answer that last question because the number of shares at any point in time is simple math, being all activity in that security from the first acquisition until "today."   The first thing I'd suggest doing is making sure that any Placeholders in the Account are properly displayed.  Do Edit > Preferences... > Investment transactions > tick "Show hidden transactions."  Then you could run an Investment transactions report for each security over the period from when the share number was correct to the time the share number turned wrong and see where the error occurred.

One possibility, and I've seen this happen, is that the "Number of shares you own of Security A" up at the broker's server doesn't get updated immediately when a transaction for Security A - Buy, Sell, whatever - occurs and you download the transaction along with what the broker (incorrectly) is indicating as your share balance.  In that instance Quicken will propose a placeholder and if you accept it or are letting Quicken automatically place transactions in your Transaction List, you'll end up with an incorrect number of shares in Quicken.

If you are allowing Quicken to automatically place transactions into the Account I'd suggest changing that and having all downloaded transactions held in the Downloaded Transactions area so that you can look at them accept them one at a time.  This will also mean that if Quicken suggests a Placeholder you can say "No" if you know that the placeholder isn't really needed.

"there are also transactions among and between accounts which need to be recorded. How can I do that"

Well, I'd think you'd know about transfers between accounts before the broker does, (I assume you either initiate each transfer or have set up some menu of transfers that the broker follows), so the best way, in my opinion, is to enter the complete transfer manually.  When the transfer occurs subsequently and is passed down to you in a download, each side of the transfer finds its "match" when the "from" or "two" Account downloads its side of the transaction.  If you don't want to do that, then when the first leg of the transfer shows up in the Downloaded Transactions window you edit it so that the other side of the transfer is properly recorded.