Are there any chapters or verses in the Book of Yehoshua that allude to Torah SheB'Al Peh?
Why was Torah SheBe'Al Peh not allowed to be written? points out that the Oral Torah was forbidden to be written down. Additionally, the prophets would only show what somebody did rather than explicate a law. Thus, as an example, we see that Elkana (the father of Shmuel Hanavi) אֶלְקָנָה בֶּן יְרֹחָם בֶּן אֱלִיהוּא בֶּן תֹּחוּ בֶן צוּף אֶפְרָתִי went on a regular basis to the mishkan in Shiloh. We also do not see a reference to the korbon Pesach after the time that Bnai Yisrael entered Eretz Yisrael even though we know it was brought. However, since there is no story that requires it to be mentioned, it is not mentioned.
Thus, anything that Yehoshua taught would have been oral and Sefer Yehoshua would not mention it unless it was needed as part of the story. While these are two separate issues, both statements are true independently.
Since you are interested in early references, lets start with an allusion at the very beginning of Joshua: Joshua 1:7-8:
רַק חֲזַק וֶאֱמַץ מְאֹד לִשְׁמֹר לַעֲשׂוֹת כְּכָל הַתּוֹרָה אֲשֶׁר צִוְּךָ מֹשֶׁה עַבְדִּי אַל תָּסוּר מִמֶּנּוּ יָמִין וּשְׂמֹאול לְמַעַן תַּשְׂכִּיל בְּכֹל אֲשֶׁר תֵּלֵךְ: לֹא יָמוּשׁ סֵפֶר הַתּוֹרָה הַזֶּה מִפִּיךָ וְהָגִיתָ בּוֹ יוֹמָם וָלַיְלָה לְמַעַן תִּשְׁמֹר לַעֲשֹוֹת כְּכָל הַכָּתוּב בּוֹ כִּי אָז תַּצְלִיחַ אֶת דְּרָכֶךָ וְאָז תַּשְׂכִּיל:
Just be strong and very courageous to observe and do in accordance with all of the Torah that Moses My servant has commanded you. Do not stray therefrom right or left, in order that you succeed wherever you go: This book of the Torah shall not leave your mouth; you shall meditate therein day and night, in order that you observe to do all that is written in it, for then will you succeed in all your ways and then will you prosper.
Verse 7 is referring to the oral Torah "that Moses my servant has commanded you." Verse 8 continues with an injunction to also study the written Torah.
The phrase in verse 7 "not straying right or left" is also clearly a reference to the discussion of the authority of the Great Court in Deutoronomy, where the exact same phrase is used, another reference to the authority of the Oral Law.