Early scientists thought that the age of the earth is proven by the genealogical tables in the book of Mathew, Mark, Luke, and John. Other than this observation, is there any reference to the direct age of the earth noted anywhere in the Bible?
No. The age of the earth is not clearly stated in the Bible.
However, according to young-earth creationists (like myself), it can be derived, assuming that the word "Day" in Genesis means a literal day, and that the genealogies are accurate.
Adding up the genealogies can give an age of about 6,000 years, but there are some gaps in the genealogies, and even among young-earth creationists, there is room for 6,000-10,000 years of history. The most literal interpretation puts the age at about 6,000 years.
Of course, there are those that hold to the day-age theory (the days in Genesis can be interpreted as "time periods") and the Gap theory (there is a gap between the creation of the universe and the arrival of man). These each throw age estimation out the window, and allow for a much older earth. Also, there are those who believe in theistic evolution, or that the Genesis account is not to be taken literally. These also throw the young earth idea out the window.
As far as other references in the Bible, again, nothing direct but Jesus did say:
Mark 10:6 (KJV) But from the beginning of the creation God made them male and female.
Young-earth Biblical literalists (such as myself) would say that this is indication that Jesus Himself considered Adam and Eve as having been created "at the beginning of creation", ruling out the gap and day-age theories. Again, there is much disagreement on this.
There is nothing in the Bible that pins down the date. Just to give an alternative view to the speculation that exists in YEC (the existing answer), the (Roman) Catholic church has an official position that is in acceptance of the scientific age of about 4.5 billion years.
I won't bother copying it all, but see Wikipedia under the heading "Pope Benedict XVI and today" (which has both the relevant quotations and sources). Furthermore, Pope Benedict XVI officially distances the (Roman) Catholic church from both YEC and ID (leaning towards theistic evolution).
I'm not Catholic myself (or even religious) but: credit where due: the Vatican knows the Bible pretty well - and they're happy to go with what the scientists say with regards to the age of the Earth.
The only starting point for all calculations are the geneologies. However, even if the age itself would be mentioned, we should not forget the following:
But, beloved, be not ignorant of this one thing, that one day is with the Lord as a thousand years, and a thousand years as one day. (2 Peter 3:8)
This can make all age calculations highly speculative, and also, irrelevant.
Archbishop Ussher believed that the age of the earth could be calculated quite accurately, using the biblical genealogies that go all the way back to Adam, whom he assumed to be created on the sixth day. Using the Bible and, to a limited extent, extra-biblical information, he decided that the world was created in 4004 BCE, a date that differs little from those of others who attempted to determine a biblical creation date. Although Ussher's chronology is hopelessly in error (scientists have established that the earth is actually 4.5 billion years old), it has been defended even by eminent scientists such as Stephen Jay Gould as representing the best of scholarship in his time.