To solve this problem, we need to find the equation of the circle and then determine the coordinates of the square's vertices.

Finding the Equation of the Circle:
The given equation of the circle is x^2 + y^2 - 2x - 4y + 3 = 0. We can rewrite this equation by completing the square for both x and y terms:

(x^2 - 2x) + (y^2 - 4y) = -3
(x^2 - 2x + 1) + (y^2 - 4y + 4) = -3 + 1 + 4
(x - 1)^2 + (y - 2)^2 = 2

From this, we can determine that the center of the circle is at (1, 2) and the radius is √2.

Determining the Coordinates of the Square's Vertices:
Since the sides of the square are parallel to the coordinate axes, the distance from the center of the circle to each side of the square will be equal to the radius (√2).

Let's assume one vertex of the square is (x, y). Since the sides of the square are parallel to the coordinate axes, the other three vertices can be determined by adding or subtracting √2 from the x and y coordinates.

Now, let's consider the given options one by one:

1) (1 + √2, -2)
If this is a vertex of the square, then the distance between this point and the center of the circle should be √2. However, if we calculate the distance using the distance formula, we get:

d = √[(1 + √2 - 1)^2 + (-2 - 2)^2]
= √[√2^2 + (-4)^2]
= √[2 + 16]
= √18

Since √18 is not equal to √2, this option is not correct.

Similarly, we can check the other options and find that none of them satisfy the condition that the distance between the vertex and the center of the circle is √2.

Therefore, the correct answer is 4) none of the above.

Note: It is important to carefully analyze each option and check if it satisfies the given conditions. In this case, the distance between the vertex and the center of the circle is a crucial criterion for determining the correct answer.