Many people might think this is not a good thing, and I partly would agree with them.
I would only agree with them if they were talking about forcing a child to learn the multiplication tables by rote memorization. I don't think introducing math concepts early is bad, you just have to explain things conceptually rather then force them on your child. Teach the concepts slowly and at your child's own pace. Some kids will get this concepts quickly, others might take a lot of time and practice to internalize the concepts. If you try and force a concept onto a child and don't explain it, you risk making something interesting and fun into something that they do not enjoy. You risk taking the wonder out of math, so be careful.
There are two prerequisites for a child to learn to multiply:
The first thing to teach is the basic concept of counting and adding groups of objects. My son picked this up awhile ago, but it is the first thing they need to know to learn multiplication. Do not start teaching multiplication without the child understanding addition and counting well. They don't have to be fast at it, but they have to know it before you begin this journey.
Second, teach counting by 2's, 5's and ten's.
Once they have these two skills down pat you can begin teaching your child multiplication.
Begin by teaching the concept of multiplication. Multiplication is just addition of number to itself a given number of times. For example: If you want to multiply 7 by 3, you add seven to itself 3 times. You should also point out that multiplying 3 by 7 will give you the same answer as well.
Teach the concept of the multiplicative identity, any number multiplied by one will yield the same number.
Teach the multiplicative property of zero, any number multiplied by zero will always yield zero. I used the analogy that zero is a black hole, and that multiplying numbers by it will result in the zero eating them up.
Create a 2 by 2 grid with the numbers 2 to 10. If you would like you can print the images I created for this purpose.
Fill in the 2's column by adding two to each of the resulting answers above it. Right after you finish the 2's column, point out that the column's contents can also be copied to the 2's row as well.
Do the same with the 5 x 5 row and column and then the 10 x 10 row and column.
All three of those rows are very easy since the child already knows to count by 2's, 5's and 10's.
Now you should teach the 'finger trick' for multiplying by nines. All the digits of all the multiples of 9 from 1 - 10 will sum together to 9.
9, 18, 27, 36, 45, 54, 63, 72, 81, 90.
Because of this fact, you can quickly multiply by nine by counting out the number you want to multiply by nine on your hands, going left to right, placing the finger down as you go. The numbers to the left of that finger will be the tens place. The number after your finger will represent the ones place. You can quickly read off the answer this fashion.
Use this technique to fill in the missing values of the 9x9 row and column.
Having done the above, we will have just a small number of grid columns that need to be filled in:
3 x 3, 3 x 4, 3 x 6, 3 x 7, 3 x 8
4 x 4, 4 x 6, 4 x 7, 4 x 8,
6 x 6, 6 x 7, 6 x 8
7 x 7, 7 x 8
And finally, 8 x 8
You have now created a multiplication table from scratch with your child, and got a lot of practice doing addition and counting along the way. Teaching multiplication in this manner focuses on the concepts of multiplication and addition, instead of just rote memorization. Once the concepts are cemented, only then is it appropriate to have them memorize and commit the tables to memory.