No there is not any blue blood in our body at any time. The reason that blood veins can appear blue is due to an “optical illusion” in which the light absorbed and reflected by the skin and blood makes the veins appear blue. In the Medieval era, nobility were referred to as “blue bloods” because their blue veins were easier to see than the peasants who worked outside all day and had tanner skin. The characteristic red color of blood is due to the complex formed between iron, oxygen and the protein hemoglobin. When the hemoglobin and iron don’t have oxygen, the blood is a deep red color and when they do have oxygen, the blood is the bright red color that most people recognize as blood. There are organisms with blue blood such as horseshoe crabs. These crabs have blue blood because instead of iron, they have copper in their blood.

When I was looking into this I actually found a similar answer on ScienceLine already.

Answer 1 is fantastic and I couldn't hope to do a better job answering.

Blood in your body is always red. There's a misconception that some blood is blue when it is de-oxygenated, or without oxygen, but it is always red - just a different shade of red. When you look at your arms, your veins seem bluish-green because the light that bounces off other tissues and skin make the blood in your veins appear that way.

Yes. Veins carry "blue" blood.

Blood contains a compound called hemoglobin. Hemoglobin is like a train, which collects people from a place and drops them in a different place. Likewise, hemoglobin collects Oxygen from the lungs and brings it to all the cells in the body. Hemoglobin in the lungs binds to oxygen forming hemoglobin+Oxygen compound, which is bright red in color. This is why the blood is red - because of oxygenated hemoglobin. The blood vessels (channels to carry blood) carrying oxygenated blood are the "arteries".

When the oxygenated hemoglobin reaches the cells, it gives up its Oxygen. Its job is not done yet. There is another transport function hemoglobin plays, which is to transport the waste product of cells, namely carbon dioxide (CO₂), to the lungs for breathing its out. To perform this, hemoglobin combines with CO₂ to form a compound that is blue in color. The blood vessels that carry hemoglobin+CO2 are called "veins". This is why veins are bluish in color. But not all CO₂ is transport through hemoglobin; some get transported as bicarbonates (salts) and some as just CO₂ gas.

This question is related to a slightly different one - Why is blood red?Inside of your red blood cells, a complex of a globular protein (globin) and a heme group form "hemoglobin." The heme group binds to iron. Iron in its oxygenated form has a reddish color - so when your blood has a lot of oxygen in it, it is bright red. Now think about your circulatory system - when blood is being pumped through the lungs, the hemoglobin is picking up lots of oxygen - and it is bright red in the arteries. As oxygen is depleted, the blood is darker red (this would be in your veins) because the iron is less oxidized. Human blood is never "blue" even though many of our veins are bluish in color - that color comes from the way light is reflected through the skin. There are other organisms whose hemolymph does have varying colors, however.