As Black, against 1. d4, what opening would you recommend me which is solid, positional, slow, strategic, quiet, closed, and similar to the Caro-Kann? I'm a 1750 Elo player. I have a very positional, quiet and strategic style of play.

As White I play 1. d4, but I plan to also learn 1. Nf3 soon. As Black against 1. e4 I play the Caro-Kann.

I would like an opening for Black against 1. d4 which meets these two conditions:


It needs to be as much similar to the Caro-Kann as possible (since I already play the Caro-Kann against 1. e4).
On average it must tend to lead to positions which are as much solid, positional, slow, closed, strategic and quiet as possible (I want nothing sharp and tactical).
It needs to be as much similar to the Caro-Kann as possible (since I already play the Caro-Kann against 1. e4).


Impossible. The problem with this approach is that you have already stopped e4 with ...d5 so White simply can not transpose even if he wished to do so. Your best bet is the Slav defense, as it is very similar to Caro-Kann ( same pawn structure for Black, with the Bf5 bishop ), but even then there are ways for White to deviate into sharp and very different lines.


  On average it must tend to lead to positions which are as much solid, positional, slow, closed, strategic and quiet as possible (I want nothing sharp and tactical).


The best candidate fitting into this description is Queen's Gambit Declined. 

Only Black can sharpen the game by transposing into Semi-Slav with Botvinnik system or with some sharp lines in the Slav with ...b5, but if you wish you can always steer the game into solid, positional lines. Then White will have no way to sharpen the game ( neither do you ), and you both will have to battle with dull, positional maneuvers and plans. 

Be warned, it is very hard to play these positions as Black and tactical skill here is practically irrelevant! Most of the time you will have a hard time defending, and you will have to learn how to properly engineer a freeing break. That is not an easy task, just so you know ( I am speaking this from experience )! 

In order to play this opening well with either color you also must know how to play typical positions very very well ( not to mention you need strong endgame knowledge! ), but this might suit your style ( based on your description in second criteria ). 

SUMMARY:

Chess would be too simple if you could play one opening against every opponent's reply. This especially applies for Black! I once too wanted to play the Caro-Kann against everything, but after very thorough study I was forced to realize that it isn't possible.

As for positional opening against 1.d4, your best bet are the dull lines of the Queen's gambit declined. In every other opening White always have a chance to sharpen the game ( always! ), but if you choose this path be ready for a long defense which will usually give you no more than a draw. Still, you will get the solidity and quietness you seek...

If you need further help or clarifications leave a comment.

Best regards.