Neither qur'an nor sunnah tell us to say :
"Sadaqa Allahu al-'Adhym"
صدق الله العظيم
It literally means:
Allah the Exalted has spoken the truth
The statement itself is correct as Allah the Almighty uses the expression "Allah spoke the truth" (or similar) when referring to Him in the qur'an:
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Say, " Allah has told the truth. So follow the religion of Abraham, inclining toward truth; and he was not of the polytheists." (3:95)
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... And who is more truthful than Allah in statement.(4:87)
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... and who is more truthful than Allah in statement.(4:122)
And the prophet () also used it in several ahadith:
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The Messenger of Allah (ﷺ) delivered a speech to us; meanwhile al-Hasan and al-Husayn came upon there stumbling, wearing red shirts. He came down from the pulpit, took them and ascended it with them. He then said: Allah truly said: "Your property and your children are only trial" (Ixiv.15). I saw both of them, and I could not wait. Afterwards he resumed the speech.
(Sunan abi Dawod)
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... The truest of word is the Book of Allah and best of guidance is the guidance of Muhammad. ...
(See for example in along hadith in Sunan an-Nasa-i)
There are no reports of the time of the Sahabah () either supporting this.
The only report I know of how the prophet () made somebody stop his recitation was saying to him (See here for example):
"Hasbuk" or "Hasbuk al-Aan"
"حسبك" or "حَسْبُكَ الْآنَ"
literally: "Enough" or "Enough for now"
Therefore scholars hold the opinion that saying this after each recitation on regular basis is bid'ah, while saying it under certain circumstances and irregularly is rather permissible (this is the most correct view). A few scholars regard it as permissible. Others consider it bid'ah.
As a Muslim you should be aware that it is not a Sunnah of the prophet () and therefore you shouldn't recommend it nor promote it as such.
There's no specific act or supplication one could perform when finishing a recitation. So you are basically free to do it in your own words or simply stop the recitation in all calm and put you Moshaf back. The report above only shows how a person who asked you to recite or read could make you stop in an adequate manner.
Historically one of the first instances in which it was quoted that we should be expressing our trust in Allah's words after reciting the qur'an is in Nawadir al-Osol fi Ahaadithi ar-Rassul نوادر الأصول في أحاديث الرسول of al-Hakim at-Tirmidhi الحكيم الترمذي (died 320 a.H.):
In the following I#m translating from Arabic langauge, as these translations are of my own take them carefully!
ومن حرمته ... إذا انتهت قراءته أن يصدق ربه، ويشهد بالبلاغ لرسوله صلى الله عليه وسلم، ويشهد على ذلك أنه حق، فيقول: صدقت رب وبلغت رسلك، ونحن على ذلك من الشاهدين، اللهم اجعلنا من شهداء الحق، القائمين بالقسط، ثم يدعو بدعوات.(Source)
Of it sacredness ... If his reading ends, he should believe his Lord, and he testifies to his Messenger, may Allah's prayers and blessings be upon him, and testify that it is true, so he says: You have believed your Lord and reached your messengers, and we are among the two witnesses, O Allah, make us among the martyrs of the truth, who make the installments, then make supplications.
So one could say that the use of this expressions was known and practiced in the 4th hijri century.
