Note: I am neither Canadian, nor a financial or legal advisor. This is neither financial nor legal advice.
TL;DR: You should be able to retrospectively apply for GST/HST, but cannot do so for Trillium.
General Filing Obligations
The document Preparing Returns for Deceased Persons 2018 from the Canada Revenue Agency (CRA) covers the general obligations that fall upon the legal representative of someone who has died. In brief:
Chapter 1 - General information
What are your responsibilities as the legal representative?
This guide deals only with your responsibilities under the Income Tax Act. Under the Act, as the legal representative, it is your responsibility to:
- file all required returns for the deceased
- ensure that all taxes owing are paid
- let the beneficiaries know which of the amounts they receive from the estate are taxable
As the legal representative, you are responsible for filing a return for the deceased for the year of death. This return is called the final return. For more information, see Chapter 2.
You also have to file any returns for previous years that the deceased person did not file. If the person did not leave records about these returns, or if you cannot tell from existing records whether or not the returns were filed, contact the CRA at 1-800-959-8281. If you have to file a return for a year before the year of death, use an Income Tax and Benefit Return for that year. Previous-year returns are available at CRA forms and publications or by calling 1-800-959-8281.
So, whether or not you may be eligible for unclaimed credits, it appears that you should file any missing returns.
On the subject of GST/HST, it first notes that any payment received after death would have to be returned:
Goods and services tax/harmonized sales tax (GST/HST) credit received after the date of death
Generally, GST/HST credit payments are issued on the 5th day of the month in July, October, January, and April. If the deceased was receiving GST/HST credit payments, the CRA may still send out a payment after the date of death because they are not aware of the death. If this happens, you should return the payment to the tax centre that serves your area.
In relation to the spouse/common-law-partner of the deceased, it then notes:
What if the deceased's GST/HST credit is for the deceased and their spouse or common-law partner?
If the deceased had a spouse or common-law partner, that person may now be eligible to receive the GST/HST credit if they filed an income tax and benefit return. The GST/HST credit payments will be based on their net income alone.
What if the surviving spouse's or common-law partner's GST/HST credit included a claim for the deceased?
If the surviving spouse's or common-law partner's GST/HST credit included an amount for the deceased, the payments will be recalculated based on the surviving spouse's or common-law partner's net income and will only include a claim for them and any eligible children, if applicable.
So, going forward the spouse/partner of the deceased may be eligible for GST/HST credits.
Claiming for Previous Years
As far as back-dated credits are concerned, the current version of the document GST/HST Credit, also from the CRA has this to say:
How do you get the credit?
To get the GST/HST credit, including any related provincial credits, you have to file a tax return for 2018, even if you have not received income in the year.
[...information for newcomers to Canada...]
You have up to 3 years to ask for a retroactive payment. If you have not filed your 2016 and 2017 tax returns within the 3-year period, you can send us a request under the taxpayer relief provisions of the Income Tax Act. For more information, go to Taxpayer relief provisions or see Information Circular IC07-1, Taxpayer Relief Provisions.
For a deceased recipient, the GST/HST credit entitlement ends the quarter following the date of the death. For more information, see Has a GST/HST credit recipient died?.
So you should be able to ask for three years of back-payments (2016/7/8 from the perspective of the above document; 2015/6/7 should have been possible at the time the original question was asked).
(Ontario) Trillium Payments
The document Ontario Trillium Benefit Questions and Answers, also from the CRA deals with Trillium (a combination of Northern Ontario Energy Credit, Ontario Energy and Property Tax Credit and Ontario Sales Tax Credit). Looking at questions 15 and 22:
15. I have not filed my 2018 income tax and benefit return. Will I still receive the OTB?
No. You must file your income tax return each year to continue to get the OTB, even if you don't have any income to report. See question 4 for information on how to apply for the OTB.
22. I forgot to apply for the OSTC, OEPTC and/or NOEC components of the OTB when I filed my prior year returns. Can I still apply?
Yes. You have to request a reassessment to the applicable return or returns. Do not file additional returns for these years. Instead, you can make your request by visiting My Account. My Account is a secure, convenient, and time saving way to access and manage your tax and benefit information online, seven days a week.
It appears you cannot make a claim for a year in which you did not file a tax-return, although you could retrospectively make a claim for a year in which you did file a return but did not apply for any of the Trillium components.