Are birthday celebrations ungodly? Friday the 20th was my birthday and as usual, I'll just have a quiet time thanking God for my life. I don't fellowship with JWs anymore but my conscience will not allow me as to why at all I should hold this 'special' day for merry making. I couldn't  devalue their view of this as ungodly.

As far as I know, there are 3 instances of birthday celebrations which all ended in killing of people. 
1st account is when Pharaoh puts his chief baker to death.


  And it came to pass the third day, which was Pharaoh’s birthday, that he made a feast unto all his servants: and he lifted up the head of the chief butler and of the chief baker among his servants...But he hanged the chief baker: as Joseph had interpreted to them.(Genesis 40:20-22).


2nd account is at Job who lost his children during their birthday feast. However and whenever they celebrated, Job knew it didn't please God.


  And his sons went and feasted in their houses, every one his day; and sent and called for their three sisters to eat and to drink with them. And it was so, when the days of their feasting were gone about, that Job sent and sanctified them, and rose up early in the morning, and offered burnt offerings according to the number of them all: for Job said, It may be that my sons have sinned, and cursed God in their hearts. Thus did Job continually. (Job 1:4-5).
  
  While he was yet speaking, there came also another, and said, Thy sons and thy daughters were eating and drinking wine in their eldest brother’s house: (Job 1:18).


From Job 3 we note that "their day" denotes birthdays. 

3rd instance is where Herod beheaded John the Baptist on his birthday only to please his daughter. 


  But when Herod’s birthday was kept, the daughter of Herodias danced before them, and pleased Herod. Whereupon he promised with an oath to give her whatsoever she would ask..."And the king was sorry: nevertheless for the oath’s sake, and them which sat with him at meat, he commanded it to be given her. And he sent, and beheaded John in the prison. (Matthew 14:6-10)


I personally think that God has shown us how to celebrate our lives keeping the true purpose in mind. ( John 10:10, John
17:17, II Tim. 3:16-17).

Today, people may not lose their lives at the celebrations and nowhere does the bible directly stop us from this activity but indirectly, what does it seem to be telling us, perhaps about the origin? All posts and counter-claims from any denomination will be helpful.
First to address a couple fuzzy or incorrect conclusions drawn in the question:

The book of Job does NOT say, "However and whenever they celebrated, Job knew it didn't please God." As you quoted, Job is being proactive to show God he cares about his children, and that on the off chance they have sinned, an atonement can be made.

Also, the beheading of John the Baptist coincided with Herod's birthday, but was not done for his birthday: it was done to please Herodias' daughter (who asked for her mother) after she apparently did some pretty exciting "dancing" for the guests at the party.

To the question itself. Firstly, a "birthday" is amoral - it is a day like any other. Days do not have morality associated with them (though the actions of celebrants/observers will have moral implications). 

Secondly, @Marc's answer is a start. Hannah celebrated Samuel's birth (though perhaps not the "day") annually:


  Now Samuel was ministering before the LORD, as a boy wearing a linen ephod. And his mother would make him a little robe and bring it to him from year to year when she would come up with her husband to offer the yearly sacrifice. Then Eli would bless Elkanah and his wife and say, “May the LORD give you children from this woman in place of the one she dedicated to the LORD.” And they went to their own home.


Paul writes in Romans 14 a generally-applicable commentary on celebrations:


  Who are you to judge the servant of another? To his own master he stands or falls; and he will stand, for the Lord is able to make him stand.
  One person regards one day above another, another regards every day alike. Each person must be fully convinced in his own mind. He who observes the day, observes it for the Lord, and he who eats, does so for the Lord, for he gives thanks to God; and he who eats not, for the Lord he does not eat, and gives thanks to God. For not one of us lives for himself, and not one dies for himself; for if we live, we live for the Lord, or if we die, we die for the Lord; therefore whether we live or die, we are the Lord’s.


There is no condemnation of celebrating certain amoral days on their own - any commentary made about varied celebrations is always about what the participant(s) may be doing (or not doing). It is not about the day itself.