What year did the Hebrew nation with Joshua cross the Jordan river into the promised land? In terms of our current and modern dating system, what year did Joshua and the Hebrew nation cross the Jordan into the land of Canaan? Also, based on this what year would the first Jubilee year have occurred as described in Leviticus 25:1-4,8,9


  When you come into the land which I give you, then the land shall keep a sabbath to the LORD. 3 Six years you shall sow your field, and six years you shall prune your vineyard, and gather its fruit; 4 but in the seventh year there shall be a sabbath of solemn rest for the land, a sabbath to the LORD. You shall neither sow your field nor prune your vineyard.’ 8 “And you shall count seven sabbaths of years for yourself, seven times seven years; and the time of the seven sabbaths of years shall be to you forty-nine years. 9 Then you shall cause the trumpet of the Jubilee to sound on the tenth day of the seventh month; on the Day of Atonement you shall make the trumpet to sound.”
The dates for these events, as with most biblical chronological estimates, are debated. Scholars cited in this answer date the crossing of the Jordan at 1406 BCE, 1260 BCE, or 1451 BCE, and the first year of Jubilee to be immediately after or some time soon after those respective dates.
Here is a (lightly edited) elucidation on the method of the calculation of the date of 1406 BCE from The Pattern of Prophecy:

We know this [date of 1406 BCE] from a key chronological marker recorded in the book of Kings. The year Solomon began to build the Temple is given both in terms of the exodus from Egypt as well as the year of Solomon’s reign:
1 Kings 6:1  And it came to pass in the 480th year after the children of Israel were come out of the land of Egypt, in the fourth year of Solomon's reign over Israel, in the month Zif, which is the second month, that he began to build the house of the LORD.
It was established in Thiele’s Mysterious Numbers of the Hebrew Kings that the fourth year of Solomon corresponds to 966 BCE.  Therefore the date of the Exodus must be 480 years prior to 966 BCE, or 1446 BCE. Since the Crossing of the Jordan took place 40 years later, that dates it to 1406 BCE.

This commenter on the Seder Olam, a traditional Rabbinic text of chronology, disagrees with that date. He also gives a date of 1268 BCE for the first year of Jubilee. Bracketed comment's are this author's, and the text has been reformatted:

When did these counts start? The Biblical text states: When you come into the land which I give you, then shall the land keep a sabbath unto the Lord. This means that the first cycle to be counted was when they will come into the land and this first count was a Sabbatical year. This first Sabbatical took place when the first tribes took possession of their inheritance, East from the Jordan River, when Moses was still alive. He gave that territory to the tribes of Reuven, Gad and half of the tribe of Manasseh (Numbers 32:33). This took place before the death of Aaron (Numbers 33:38-39) in Hebrew year 2494 [1266 BCE].
After the conquest, the land of Canaan was also divided and given as inheritance to the other tribes. The final inheritance was given to Caleb from the tribe of Judah after he made a plea to Joshua: Caleb was then 85 years old. He must have made his plea just after the end of the first Jubilee, so that a count of 50 years will be fully available to him and his tribe. Therefore the First Jubilee was counted when Caleb was 84 years old. This was in Hebrew year 2500. And the first Sabbatical year was counted 8 years earlier (1 year as the Jubilee year + 7 years for a full Sabbatical cycle), therefore this was in Hebrew year 2500-8= 2492. This indeed was at a time when Aaron was still alive and he performed the first celebrated Sabbatical year. But both he and Moses died before they could celebrate the first Jubilee year in 2500 [which corresponds to 1260 BCE].

In a more Scholarly approach in his book Old Testament Chronology: A Return to the Basics (pdf), Dr. Floyd Nolan Jones gives yet a different date for the crossing of the Jordan supported by previous scholars, and also identifies the first year of Jubilee as 1395 BCE (parenthetical notes are Jones'):

The year Moses died and Joshua entered the land was long ago determined by Ussher, Bishop Loyd, Nicholas Toinard, William Wiston, and now independently confirmed by this author [Jones] as 1451 BC. However, the year of Jubilee did not have to do with merely being in the land but  with its actual possession and cultivation (Lev. 25).
After seven years of conflict with the Canaanites, the wars ended at the close of 1445 BC (c. April 1451 - 1445 = 7 years inclusive). From the base camp at Gilgal, Joshua then gave the tribes of Judah and Joseph their portions. Early in 1444, the tabernacle was moved to Shiloh. The rest of the land west of the Jordan was then divided among the remaining seven tribes, and the men from the two-and-a-half tribes east of the Jordan returned home.
Until this, Israel had lived off of the crops of the Canaanites, volunteer crops and supplies from the eastern two-and-a-half tribes (Josh. 24:13). Israel's tillage thus began in 1444. Hence, fom this year are reckoned the Sabbatic and Jubilee years (1444- 49 = 1395 BC, the first Jubilee. See p. 289 ff).